Contributions to the theory of normal affine semigroup rings and Ulrich modules of rank one over determinantal rings [Elektronische Ressource] / von Attila Wiebe

Prof.Dr. Uwe Storch (Bochum #1) 1236 2001 May 30 14:22:22

Description

Contributions to the theory ofnormal affine semigroup ringsandUlrich modules of rank oneover determinantal ringsDissertationdem Fachbereich Mathematikder Universit¨ at Duisburg-Essenzur Erlangung desDoktorgrades (Dr. rer. nat)vorgelegt im April 2006vonAttila Wiebeaus BochumTag der mundlic¨ hen Prufung:¨ 30. Juni 2006Vorsitzender der Prufungsk¨ ommission: Prof. Dr. Rudiger¨ G¨ obelGutachter: Prof. Dr. Winfried BrunsProf. Dr. Jurgen¨ HerzogFor my wife1ContentsIntroduction 21. The Rees algebra of a normal affine semigroup ring 41.1. Affine semigroup rings 41.2. A short excursion into convex geometry 61.3. The bottom of an affine semigroup 91.4. The integral closure of an ideal 101.5. The associated graded ring of an affine semigroup ring 121.6. Normality of the Rees algebra 141.7. Cohen-Macaulayness of the Rees algebra 211.8. The special case of hypersurface rings 262. On the type of a simplicial normal affine semigroup ring 302.1. Preparations 302.2. The case of dimension 2 322.3. The case of 3 323. Ulrich modules of rank one over determinantal rings 383.1. Ulrich modules 383.2. Determinantal rings 413.3. The existence of Ulrich modules of rank one 43References 522IntroductionThe theory of affine semigroup rings and the theory of determinantal rings areappealing and vital branches of present-day commutative algebra. In the investi-gation of these rings, both geometric and combinatoric aspects play an importantrole.

Sujets

• Mathematics

Informations

• Publié par: universitat_duisburg-essen
• Publié le: 01 janvier 2006
• Nombre de lectures: 7
• Langue: English

Extrait

Contributions to the theory of normal affine semigroup rings and Ulrich modules of rank one over determinantal rings

Dissertation

dem Fachbereich Mathematik derUniversit¨atDuisburg-Essen zur Erlangung des Doktorgrades (Dr. rer. nat)

vorgelegt im April 2006 von Attila Wiebe aus Bochum

Tagderm¨undlichenPr¨ufung:30.Juni2006 VorsitzenderderPr¨ufungskommission:Prof.Dr.R¨udigerGo¨bel Gutachter: Prof. Dr. Winfried Bruns Prof.Dr.J¨urgenHerzog

For

my

wife

Contents

Introduction 1. The Rees algebra of a normal affine semigroup ring 1.1. Affine semigroup rings 1.2. A short excursion into convex geometry 1.3. The bottom of an affine semigroup 1.4. The integral closure of an ideal 1.5. The associated graded ring of an affine semigroup ring 1.6. Normality of the Rees algebra 1.7. Cohen-Macaulayness of the Rees algebra 1.8. The special case of hypersurface rings 2. On the type of a simplicial normal affine semigroup ring 2.1. Preparations 2.2. The case of dimension 2 2.3. The case of dimension 3 3. Ulrich modules of rank one over determinantal rings 3.1. Ulrich modules 3.2. Determinantal rings 3.3. The existence of Ulrich modules of rank one References

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2

Introduction

The theory of affine semigroup rings and the theory of determinantal rings are appealing and vital branches of present-day commutative algebra. In the investi- gation of these rings, both geometric and combinatoric aspects play an important role. An affine semigroup ringRis a finitely generated algebra over a fieldK, which is isomorphic to aZ n -graded subalgebra of the ringK[x 1 ±1 , . . . , x n ±1 ] of Laurent poly- nomials. If the unit groupR × is equal toK × , we callRa positive affine semigroup ring. A fundamental result of affine semigroup rings is Hochster’s theorem, which states that a normal affine semigroup ring is Cohen-Macaulay. In our thesis, we will mainly consider positive normal affine semigroup rings. Normal affine semigroup rings occur for example in invariant theory: ifKis an algebraically closed field andTis a torus group overKwhich acts linearly on A=K[x 1 ±1 , . . . , x n ±1 ], then the ringA T of invariants is a normal affine semigroup ring. Therefore, some authors use the term ‘toric ring’ instead of ‘normal affine semigroup ring’. The general definition of a determinantal ring is rather complicated. However, in this thesis, we only consider determinantal rings of the formK[X]/I r+1 (X), where Xis anm×n-matrix of indeterminates over a fieldK, andI r+1 (X) denotes the ideal ofK[X] which is generated by the (r of+ 1)-minorsX. Determinantal rings are the most prominent example of algebras with straight- ening law. Just as normal affine semigroup rings, determinantal rings are normal Cohen-Macaulay domains. In the first section, we study the Rees algebra of a positive normal affine semigroup ringRrespect to its graded maximal idealwith m is obvious that. ItR[mt] is again a positive affine semigroup ring. But in general,R[mt] is not normal. fact, we In show thatR[mt] may even fail to be Cohen-Macaulay. The main result of the first section is a normality criterion for the Rees algebra: we prove thatR[mt] is normal if and only if the powersm i , i= 1, . . . , d−2, with d= dimR, are integrally closed inR. As a corollary, we obtain thatR[mt] is normal if dimR≤3. When proving the normality criterion, we make use of some notions from convex geometry that we learned from the preprint [BG2] of Bruns and Gubeladze. Also, the monographs [Va1] and [Va2] of Vasconcelos were valuable sources of inspiration when writing this section. A large part of this section is contained in the author’s article [Wi], which will be published soon in Manuscripta Mathematica. The second section is devoted to the typer(R) of a simplicial normal affine semi- group ringRof dimensiond≤ type (some authors say: Cohen-Macaulay3. The type) is an important numerical invariant ofR is equal to the minimal number. It of generators of the canonical module ofR in a sense, it measures how. Therefore, farRis away from being Gorenstein. We prove thatr(R) is bounded above byr( P), wherePis the special fibre of an embeddingR →P:=K[x 1 , . . . , x d ].

3 In the third section, we turn to determinantal rings. We show that the divisor class group of a determinantal ringR=K[X]/I r+1 (X) contains two outstanding classes: the ideals which represent these classes are Ulrich modules of rank one. Although affine semigroup rings seem to have little to do with this subject, they play a crucial role in the proof of the main theorem of that section. The results of this section appeared in the joint paper [BRW] with Bruns and R¨omer. Terminology We say that a domain is normal, if it is Noetherian and integrally closed in its field of fractions. A finitely generated graded algebraAover a fieldKis called standard graded, ifA 0 =KandA=K[A 1 ]. The symbolNdenotes the set of all positive integers, andN 0 denotesN∪ {0}. The symbolQ + (resp.R + ) denotes the set of all nonnegative rational (resp. real) numbers. We use⊂for a proper inclusion and⊆to mean “contained in or equal to”. Acknowledgements I wish to express my gratitude to my advisor Professor J¨urgen Herzog for his warm-hearted support and for his unresting willingness to discuss all my questions and problems. Also, I am deeply indebted to my wife Barbara for her continuous encouragement and moral support during the preparation of this thesis.